Final answer:
The wavelength of an alpha particle accelerated through a potential difference of 100 V can be calculated using the equation λ = h/mv, where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is the velocity of the particle. The alpha particle has a mass of 1.67 x 10^-27 kg and a velocity of 4.66 x 10^6 m/s, resulting in a wavelength of 8.68 x 10^-15 m.
Step-by-step explanation:
The wavelength of a particle can be calculated using the equation λ = h/mv, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 J s), m is the mass of the particle, and v is the velocity of the particle.
In this case, the particle is an alpha particle with a mass of m_P = 1.67 x 10^-27 kg.
The alpha particle is accelerated through a potential difference of 100 V.
To find the velocity of the alpha particle, we can equate the kinetic energy gained from the potential difference to the initial rest energy of the particle.
E_kinetic = q * V
= (1.6 x 10^-19 C) * (100 V)
= 1.6 x 10^-17 J
E_rest = m_P * c^2
= (1.67 x 10^-27 kg) * (3 x 10^8 m/s)^2
= 1.503 x 10^-10 J
The kinetic energy gained is E_kinetic = E_rest,
so we can equate the two and solve for the velocity, v:
v = sqrt(2 * E_kinetic / m_P)
= sqrt(2 * 1.6 x 10^-17 J / 1.67 x 10^-27 kg)
= 4.66 x 10^6 m/s
Now, using the equation λ = h / (m_P * v), we can calculate the wavelength:
λ = (6.626 x 10^-34 J s) / (1.67 x 10^-27 kg * 4.66 x 10^6 m/s)
= 8.68 x 10^-15 m
Therefore, the wavelength of the alpha particle accelerated through a potential difference of 100 V is 8.68 x 10^-15 m.